Tree cumulants and the geometry of binary tree models

TL;DR

This paper introduces a new parametrization for binary tree models using tree cumulants, providing insights into model identifiability and the geometry of the parameter space, with explicit formulas when identifiable.

Contribution

The paper develops a novel cumulant-based parametrization for binary tree models, clarifies identifiability conditions, and describes the geometry of the unidentified parameter space.

Findings

Provides necessary and sufficient conditions for model identifiability.

Offers explicit formulas for model parameters when identifiable.

Describes the geometry of the parameter space in non-identifiable cases.

Abstract

In this paper we investigate undirected discrete graphical tree models when all the variables in the system are binary, where leaves represent the observable variables and where all the inner nodes are unobserved. A novel approach based on the theory of partially ordered sets allows us to obtain a convenient parametrization of this model class. The construction of the proposed coordinate system mirrors the combinatorial definition of cumulants. A simple product-like form of the resulting parametrization gives insight into identifiability issues associated with this model class. In particular, we provide necessary and sufficient conditions for such a model to be identified up to the switching of labels of the inner nodes. When these conditions hold, we give explicit formulas for the parameters of the model. Whenever the model fails to be identified, we use the new parametrization to describe the geometry of the unidentified parameter space. We illustrate these results using a simple example.